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Wavelet Analysis: Concepts

Wavelets decompose a signal into approximations and details at different scales, making them useful for applications such as data compression, detecting features, and removing noise from signals. This course covers continuous, discrete, and stationary wavelet transforms, as well as wavelet algorithms and visualizations.

Level: Intermediate

A basic familiarity with transforms and data smoothing is recommended for this course.
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Outline

  • Introduction to Wavelets
    Brief overview of Fourier analysis and Fourier transforms, introduction to wavelets, scaling and translation of wavelets, comparisons between Fourier and wavelet analysis
  • Continuous Wavelet Transforms
    Wavelet families; ContinuousWaveletTransform; ContinuousWaveletData; InverseContinuousWaveletTransform; working with wavelet coefficients; WaveletMapIndexed; applications including feature detection, speech analysis, and filtering signals
  • Discrete Wavelets
    Wavelet filter bank, lowpass/highpass filters, scaling and wavelet functions, parametrization and multivariate extensions of scaling and wavelet functions
  • Discrete Wavelet Transforms
    Basic concepts; standard and packet transforms; DiscreteWaveletData; working with wavelet coefficients; applications including working with symbolic arrays, financial series trends, and three-dimensional wavelet transform
  • Stationary Wavelet Transforms
    Basic concepts, standard and packet transforms, applications including numerical differentiation and image fusion
  • Wavelet Visualizations
    Plot coefficient trees; discrete and continous scalograms; plot matrix coefficients; plot image coefficients: WaveletScalogram, WaveletListPlot, WaveletMatrixPlot, and WaveletImagePlot
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